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A280317 Ordinate of points (x,y) of the square lattice such that x >= 0 and 0 <= y <= x, and ranked in order of increasing distance from the origin. Equidistant points are ranked in order of increasing ordinate.

%I A280317 #13 Jul 20 2017 08:33:22
%S A280317 0,0,1,0,1,2,0,1,2,0,1,3,2,0,3,1,2,4,3,0,1,2,4,3,0,1,5,4,2,3,5,0,1,4,
%T A280317 2,6,3,5,4,0,1,2,6,5,3,4,7,0,6,1,2,5,3,7,4,6,0,1,2,5,8,3,7,6,4,0,1,8,
%U A280317 5,2,7,3,6,4,9,8,0,5,1,7,2,3,6,9,4,8,7,5,0,1,2,10,9,3,6,8,4,7,5,10
%N A280317 Ordinate of points (x,y) of the square lattice such that x >= 0 and 0 <= y <= x, and ranked in order of increasing distance from the origin. Equidistant points are ranked in order of increasing ordinate.
%e A280317 a(12) = 3 since the twelfth point in distance from the origin is (3,3) at a distance of 3*sqrt(2) = 4.242640... whereas the eleventh is (4,1) at a distance of sqrt(17) = 4.12310... and the thirteenth is (4,2) at a distance of 2*sqrt(5) = 4.472113... .
%e A280317 The fourteenth and fifteenth points are respectively (5,0) and (4,3) and have the same distance 5 to the origin, but (5,0) has a smaller ordinate than (4,3), so a(14) = 0 and a(15) = 3.
%t A280317 xmax = 20; (* Maximum explorative abscissa *)
%t A280317 (* t are points in the triangle of vertices (0, 0), (0, max) and (xmax, xmax) *)
%t A280317 t = Flatten[Table[{x, y}, {x, 0, xmax}, {y, 0, x}], 1];
%t A280317 nmax = Floor[xmax^2/4] (* Safe limit for correctly sorted sequence *)
%t A280317 Transpose[SortBy[t, {#[[1]]^2 + #[[2]]^2 &, #[[2]] &}]][[2]][[1 ;;
%t A280317    nmax]]
%Y A280317 Cf. A280079.
%K A280317 nonn
%O A280317 1,6
%A A280317 _Robert G. Wilson v_ and _Andres Cicuttin_, Dec 31 2016